Comments for MEDB 5502, Week 04

Topics to be covered

What you will learn
- Indicator variables for three or more categories
- Multiple factor analysis of variance
- Checking assumptions of analysis of variance
- Interactions in analysis of variance
- Interactions in analysis of covariance
- Interactions in multiple linear regression
- Unbalanced data

Review oneway analysis of variance

\(H_0:\ \mu_1=\mu_2=...=\mu_k\)
\(H_1:\ \mu_i \ne \mu_j\) for some i, j
- Reject \(H_0\) if F-ratio is large
Note: when k=2, use analysis of variance or t-test

Full moon data

Admission rates to mental health clinic before, during, and after full moon.
One year of data

Boxplot of full moon data

Descriptive statistics

Analysis of variance table

Tukey post hoc

Creating indicator variables

Running general linear model with all indicator variables

Analysis of variance table with first and second indicators

Irrelevant rows removed

Parameter estimates, 1 of 3

11.458 - 13.417 = -1.959
10.917 - 13.417 = -2.5

Here are the parameter estimates. The intercept is the estimated average rate when the first and second indicator both equal zero. This is effectively the average rate for the third group (during a full moon).

The first slope is the estimated average difference in rates when you increase the first indicator by one unit and hold the second indicator constant. This is effectively the the average rate for the first group (after a full moon) minus the average rate for the third group (during a full moon). When the days switch from during a full moon to after a full moon, the estimated average admission rate decreases by almost two patients per day.

The second slope is the estimated average difference in rates when you increase the second indicator by one unit and hold the first indicator constant. This is effectively the the average rate for the second group (before a full moon) minus the average rate for the third group (during a full moon). When the days switch from during a full moon to before a full moon, the estimated average admission rate decreases by 2.5 patients per day.

Parameter estimates, 2 of 3

11.458 - 10.917 = 0.541
13.417 - 10.917 = 2.5

Parameter estimates, 3 of 3

10.917 - 11.458 = -0.541
13.417 - 11.458 = 1.959
\(\ \)
Reference category, the category associated with the indicator variable left out of the model.

You have a third choice: leaving out the first indicator. The intercept now represents the estimated average for the first group (before a full moon) which is about 11.5 patients per day. The first slope tells you that you see about half a patient less on average after a full moon compared to before a full moon. The second slope tells you that you see about 2 patients more on average during a full moon compared to before a full moon.

You may see the term “reference category”. This represents the category associated with the left out indicator variable. The other categories are compared to the reference category.

Which is the best interpretation? It depends a lot on the context of the problem. I would argue that comparing before and after to during makes the most sense, but you could argue for any of these three interpretations.

Or you could synthesize these. Average admission rates are 11.5 patients per day before a full moon, increase by about 2.5 patients per day on average during a full moon and drop by about 2 patients per day after a full moon. Then admission rates jump back up by about half a patient per day when you switch back to days before a full moon.

Using moon as a fixed factor

Removing the unneeded rows

Parameter estimates using Moon as a fixed factor

Live demo, Multiple factor analysis of variance

Break #1

What you have learned
- Indicator variables for three or more categories
What’s coming next
- Multiple factor analysis of variance

Mathematical model

\(Y_{ijk} = \mu + \alpha_i + \beta_j +\epsilon_{ijk}\)
- i=1,…,a levels of the first categorical variable
- j=1,…,b levels of the second categorical variable
- k=1,…,n replicates with first and second categories

\(\ \)

\(H_0:\ \alpha_i=0\) for all i

\(\ \)

\(H_0:\ \beta_j=0\) for all j

Crosstabulation of categorical predictors

Analysis of variance table for moon data

Removing irrelevant rows

Parameter estimates for the full moon model

Tukey post hoc test

Live demo, Multiple factor analysis of variance

Break #2

What you have learned
- Multiple factor analysis of variance
What’s coming next
- Checking assumptions of analysis of variance

Assumptions

Normality
Equal variances
Independence
Note: No linearity assumption
- Only for linear regression and analysis of covariance

Q-Q plot of residuals

Residual versus predicted value plot

Live demo, Checking assumptions of analysis of variance

Break #3

What you have learned
- Checking assumptions of analysis of variance
What’s coming next
- Interactions in analysis of variance

What is an interaction

Impact of one variable is influenced by a second variable
Example, influence of alcohol on sleeping pills
Three types of interactions
- Between two categorical predictors
- Between a categorical and a continuous predictor
- Between two continuous predictors
Interactions greatly complicate interpretation

Interaction plot

X axis, first categorical variable
Separate lines for second categorical variable
Y axis, average outcome

Hypothetical interaction plots, 1 of 4

No interaction
Ineffective treatment
Boys/girls similar

No interaction
Ineffective treatment
Boys fare better than girls

An interaction plot shows the mean values for each of the two categories. In this example, there is a placebo and a treatment. The outcome is unspecified, but a larger value is presumed to represent a better outcome. This is a pediatric example and the data is subdivided into two populations, boys and girls.

The flatness or steepness of the lines indicates whether patients given the treatment fare better than patients given the placebo.

The separation (if there is any) between the lines measures whether boys fare better or worse than girls.

If the lines have roughly the same slope (both are flat or both are steep), then there is no interaction.

In the plot on the left, the two lines are flat, indicating that the treatment is ineffective. The outcome is not changed from the placebo.

The two lines lie more or less on top of one another. This indicates that there is no difference in average outcome between boys are girls.

In the plot on the right, the two lines are flat. The treatment is ineffective. There is, however, a difference. The average outcome for boys is a lot better both in the placebo group and the treatment group. The lines are roughly parallel, indicating no interaction.

Hypothetical interaction plots, 2 of 4

No interaction
Effective treatment
Boys/girls similar

No interaction
Effective treatment
Boys fare better than girls

Hypothetical interaction plots, 3 of 4

Significant interaction
Harmful treatment in boys
Effective treatment in girls

Significant interaction
Ineffective treatment in boys
Effective treatment in girls

Hypothetical interaction plots, 4 of 4

Significant interaction
Girls fare better overall
Effective treatment
Much more effective in boys

Data dictionary for exercise data, 1 of 3

data_dictionary: exercise.sas7bdat

description: |
  This dataset is used in a tutorial about interactions. A description from the original source: The dataset consists of data describing the amount of weight loss achieved by 900 participants in a year-long study of 3 different exercise programs, a jogging program, a swimming program, and a reading program which serves as a control activity. Researchers were interested in how the weekly number of hours subjects chose to exercise predicted weight loss.

Data dictionary for exercise data, 2 of 3

loss:
  label: Average weekly weight loss
  note: negative scores denote weight gain
  scale: real
  unit: unknown, presumably pounds

hours:
  label: weekly average amount of exercise
  unit: hours

effort:
  label: weekly effort scores
  note: self report
  scale: non-negative integer
  range: 0 through 50
  direction: 50 denoting maximum physical effort

Data dictionary for exercise data, 3 of 3

prog:
  label: exercise program
  note: reading is a control
  values:
    Jogging: 1
    Swimming: 2
    Reading: 3

female:
  scale: binary categorical
  values:
    male: 0
    female: 1

satisfied:
  label: satisfied with weight lost
  scale: binary categorical
  values:
    Dissatisfied: 0
    Satisfied: 1

Box plots of exercise data

Mean values for the interaction

Analysis of variance table for interaction model

Parameter estimates for the interaction model

The intercept is the estimated average weight loss when all the indicator variables are equal to zero. That means the two reference categories: females for gender and reading for the exercise group. If you join a book club expect to gain more than three pounds.

The estimate for female=0 is the how much different the weight loss is in the reading group when you change gender from female to male. It’s a small change and not statistically significant.

The estimate for group=1 measures how much difference you see in the average weight loss in the jogging program compared to the reading program in the reference category (female). There is almost an 8 pound difference and this is statistically significant.

The estimate for group=2 measures how much difference you see in the average weight loss in the swimming program compared to the reading program in the reference category (female). There is an astounding 32 plus pound difference between swimming and reading, but again for the reference category of females.

The estimate for female=0*prog=1 shows how much different the benefit of jogging over reading is for men compared to women. Men did pretty well on the jogging program, losing 11 pounds instead of gaining more than 3 pounds. This is more impressive than for women who lost 4 pounds instead of gaining 3.

The estimate for female=0*prog=2 shows how much different the benefit of swimming is for men than for women. Both groups are better off swimming than reading, men losing 22 pounds versus a 3 pound gain. But the falls short of the benefit of swimming over reading for women a 29 pound loss versus a 3 pound gain.

Both of the parameters associated with the interaction are statistically significant.

Bottom line is that the benefits of jogging over reading and the benefits of swimming over reading are not the same for men and women.

Interaction plot, 1 of 2

An interaction plot draws a multiple lines connecting means. In this graph the top line shows the mean weight loss in the swimming program with the men’s mean on the left and the women’s mean on the right. The middle line shows the mean weight loss in the jogging program, again with the men’s mean on the left and the women’s on the right. The bottom line shows the mean weight loss (actually a weight gain!) in the reading program. Clearly the swimming program is the best, a bit better for women than men because the line slopes upward. The jogging program is second best, a bit worse for women than for men. The reading program is worst and the benefit, if you could call it that being about the same for men and women (the lines are flat).

It is a lack of parallelism that is the hallmark of an interaction.

Interaction plot, 2 of 2

When you can’t estimate an interaction

Special case, n=1
- Only one observation for categorical combination

Example, full moon study, 1 of 2

Example, full moon study, 2 of 2

Live demo, Interactions in analysis of variance

Break #4

What you have learned
- Interactions in analysis of variance
What’s coming next
- Interactions in analysis of covariance

A second type of interaction

Interactions in analysis of covariance
- Between categorical predictor and continuous predictor
- Different slopes within each category

Interaction between exercise program and hours spent exercising

Testing for interaction in analysis of covariance

Table with irrelevant rows removed

Parameter estimates

Intercept for prog=1, -8.997 + 2.216 = -6.781
Intercept for prog=2, 9.993 + 2.216 = 12.209
Intercept for prog=3, 2.216
Slope for prog=1, 10.409 + -2.956 = 7.453
Slope for prog=2, 9.83 + -2.956 = 6.874
Slope for prog=3, -2.956

The first three coefficients are a bit of an extrapolation because they represent patients who spend zero hours on the proposed intervention.

The intercept is the estimated average weight loss in the reference category (group=3 or reading program) when the number of hours devoted to the program is zero. In this case it is a weight gain of 2 pounds.

The coefficient for group=1 is how much better the weight loss is when you switch from the reading program to the jogging program, but still put in zero hours.

The coefficient for group=2 is how much better the weight loss is on average when you switch from the reading program to the jogging program, but still put in zero hours.

The coefficient for hours is how much better the weight loss is when you add an extra hour of effort and you are in the reading program. Actually, it is how much worse. Each extra hour of reading and you gain an extra three pounds on average.

The coefficient for hours*group=1 shows the change in slope when you switch from the reading program to the jogging program. Each hour invested is 10 pounds better (a loss of 7 pounds per hour invested instead of a gain of 3 pounds per hour invested) when you switch from the reading program to the jogging program.

The interpretation for hours*group=2 is similar It shows the change in slope when you switch from the reading program to the swimming program. Each hour invested is 10 pounds better (a loss of 7 pounds per hour invested instead of a gain of 3 pounds per hour invested) when you switch from the reading program to the swimming program.

Live demo, Interactions in analysis of covariance

Break #5

What you have learned
- Interactions in analysis of covariance
What’s coming next
- Interactions in multiple linear regression

Analysis of variance table

Table of means

Centered analysis

Weight loss at various conditions

hours = 2 (mean), effort = 30 (mean),
- \(\hat Y\) = 10.005
hours = 4 (mean+2), effort = 30 (mean),
- \(\hat y\) = 10.005 + 2.291*2 = 14.587
hours = 2 (mean), effort = 40 (mean+20)
- \(\hat Y\) = 10.005 + 0.707*20 = 24.145
hours = 4 (mean+2), effort = 40 (mean+20)
- \(\hat Y\) = 10.005 + 2.291*2 + 0.707*20 + 0.393*2*20 = 44.447

Live demo, Interactions in multiple linear regression

Break #6

What you have learned
- Interactions in multiple linear regression
What’s coming next
- Unbalanced data

FEV data

Descriptive Abstract: Sample of 654 youths, aged 3 to 19, in the area of East Boston during middle to late 1970’s. Interest concerns the relationship between smoking and FEV. Since the study is necessarily observational, statistical adjustment via regression models clarifies the relationship.

fev continuous measure (liters)
sex discrete/nominal (Female coded 0, Male coded 1)
smoke discrete/nominal (Nonsmoker coded 0, Smoker coded 1)
- Source: https://jse.amstat.org/datasets/fev.txt

Line plots of means for unbalanced data

Table of means

Table of frequencies and column percentages

Live demo, Unbalanced data

Summary

What you have learned
- Indicator variables for three or more categories
- Multiple factor analysis of variance
- Checking assumptions of analysis of variance
- Interactions in analysis of variance
- Interactions in analysis of covariance
- Interactions in multiple linear regression
- Unbalanced data

Additional topics??

Learning objectives for multifactor ANOVA
- Define Factor
- Define Level
- Assess types of ANOVA based on factors and levels
- Calculate the maximum number of groups in a multi-factorial ANOVA
- Define interaction
- Explain how to determine if an interaction between the factors exists
- Explain how to “name” a multi-factorial ANOVA
- Write the Null and Alternative Hypothesis for a multi-factorial ANOVA
- Write the Research Questions for a multi-factorial ANOVA
- Write the Research Interaction Questions for a multi-factorial ANOVA
- Define and explain the order of interpretation for a multi-factorial ANOVA
- Define Main Effects
- Explain the test of the Main Effects
- Define Simple Main Effects
- Explain the test of the Simple Main Effects
- Define Interaction Effect
- Explain the test of the Interaction Effect
- Explain what information the Interaction Effect test provides
- Explain the difference in information obtained between a 2-way ANOVA and 2 one way ANOVA’s
- Define the “Differences Between the Differences”
- Calculate the “Differences Between the Differences”
- Draw conclusions about the “Differences Between the Differences”
- Interpret the Interaction Effect test
- Determine the next steps of interpretation with a significant interaction effect
- Determine the next steps of interpretation with a non-significant interaction effect
- Interpret the Main Effects
- Interpret the Simple Main Effects
- List and define the Assumptions and tests for a multi-factorial ANOVA
- Define Marginal Means
- Explain the difference between Marginal Means and Means
- Complete a statistical write-up of a multi-factorial ANOVA with all elements included